![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ring2idlqus | Structured version Visualization version GIF version |
Description: For every unital ring there is a (two-sided) ideal of the ring (in fact the base set of the ring itself) which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 13-Feb-2025.) |
Ref | Expression |
---|---|
ring2idlqus | ⊢ (𝑅 ∈ Ring → ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
2 | eqid 2731 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | 2idl1 21017 | . 2 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (2Ideal‘𝑅)) |
4 | oveq2 7420 | . . . . 5 ⊢ (𝑖 = (Base‘𝑅) → (𝑅 ↾s 𝑖) = (𝑅 ↾s (Base‘𝑅))) | |
5 | 4 | eleq1d 2817 | . . . 4 ⊢ (𝑖 = (Base‘𝑅) → ((𝑅 ↾s 𝑖) ∈ Ring ↔ (𝑅 ↾s (Base‘𝑅)) ∈ Ring)) |
6 | oveq2 7420 | . . . . . 6 ⊢ (𝑖 = (Base‘𝑅) → (𝑅 ~QG 𝑖) = (𝑅 ~QG (Base‘𝑅))) | |
7 | 6 | oveq2d 7428 | . . . . 5 ⊢ (𝑖 = (Base‘𝑅) → (𝑅 /s (𝑅 ~QG 𝑖)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))) |
8 | 7 | eleq1d 2817 | . . . 4 ⊢ (𝑖 = (Base‘𝑅) → ((𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring ↔ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) ∈ Ring)) |
9 | 5, 8 | anbi12d 630 | . . 3 ⊢ (𝑖 = (Base‘𝑅) → (((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring) ↔ ((𝑅 ↾s (Base‘𝑅)) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) ∈ Ring))) |
10 | 9 | adantl 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 = (Base‘𝑅)) → (((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring) ↔ ((𝑅 ↾s (Base‘𝑅)) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) ∈ Ring))) |
11 | 2 | subrgid 20467 | . . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
12 | eqid 2731 | . . . . 5 ⊢ (𝑅 ↾s (Base‘𝑅)) = (𝑅 ↾s (Base‘𝑅)) | |
13 | 12 | subrgring 20468 | . . . 4 ⊢ ((Base‘𝑅) ∈ (SubRing‘𝑅) → (𝑅 ↾s (Base‘𝑅)) ∈ Ring) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑅 ↾s (Base‘𝑅)) ∈ Ring) |
15 | eqid 2731 | . . . . 5 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))) | |
16 | 15, 1 | qusring 21027 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (Base‘𝑅) ∈ (2Ideal‘𝑅)) → (𝑅 /s (𝑅 ~QG (Base‘𝑅))) ∈ Ring) |
17 | 3, 16 | mpdan 684 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑅 /s (𝑅 ~QG (Base‘𝑅))) ∈ Ring) |
18 | 14, 17 | jca 511 | . 2 ⊢ (𝑅 ∈ Ring → ((𝑅 ↾s (Base‘𝑅)) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) ∈ Ring)) |
19 | 3, 10, 18 | rspcedvd 3614 | 1 ⊢ (𝑅 ∈ Ring → ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾s cress 17180 /s cqus 17458 ~QG cqg 19042 Ringcrg 20131 SubRingcsubrg 20461 2Idealc2idl 21009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-ec 8711 df-qs 8715 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-0g 17394 df-imas 17461 df-qus 17462 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-minusg 18862 df-sbg 18863 df-subg 19043 df-nsg 19044 df-eqg 19045 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-oppr 20229 df-subrg 20463 df-lmod 20620 df-lss 20691 df-sra 20934 df-rgmod 20935 df-lidl 20936 df-2idl 21010 |
This theorem is referenced by: ring2idlqusb 21073 |
Copyright terms: Public domain | W3C validator |